VOLUMES OF SOLIDS AS RELATED TO TRANSVERSE SECTIONS. 57 



It is obvious that an infinite number of such expressions may be 

 obtained, satisfying any two indices. A little consideration will 

 shew that a definite system of weight-coefficients deduced to 

 satisfy two indices will in general satisfy a third conjugate index. 

 Suppose for example we put in (42) the values a = - L % ; /3 = 1 ; 

 and y = 3V, viz., those satisfying p = 2, q = 4 in Table VII., 

 when b = J : if then we calculate the values for b corresponding 

 to different values of p, we shall find that for p = ■£§■ b = I, and 

 for p - 54- b = 0. Between p = 2 and p = 4 b is greater than J; 

 and between p — about § and p = 2 less than J : — see Curve 4, 

 on Fig. 1, which exhibits the whole curve between the indicated 

 limits. Hence, with the coefficients adopted, the function 



A z = A + Bz u + Cz y + Dz w 

 would have been satisfied, u, v, and w being the three conjugate 

 indices. 



From the figure referred to — Curve 4, Fig. 1 — it is evident 

 that u and v, or v and w, may become identical for a particular 

 value of b : so also for particular weights the whole three may 

 become identical. If, for example the coefficient be unity for each 

 of the three sections, and the position of b is alone to be deter- 

 mined, we shall have from (5) or from (42a) 



b" =-^ (44) 



1 + p 



Since b can neither be greater than unity, nor negative, the limits 

 of p are \ and 2, the corresponding limits of b being 1 and ; 

 hence no values outside these can be satisfied. Further since within 

 the limits there is one and only one value of b corresponding to 

 any definite value of p, and vice versa, only one value of p can be 

 satisfied. Hence the function in such a case is reduced to 



A z = A-+Bz*; 2>p>±; 

 and the formula for area or volume to 



V= ±z(A + B 1+ C ) (45) 



The following table contains sixteen values of b between the 

 indicated limits. 



